Efficient Newton-multigrid FEM Solver for Multifield Nonlinear Coupled Problems Applied to Thixoviscoplastic Flows

This note is concerned with efficient Newton-multigrid FEM solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two-field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two-way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. [1, 2]. We continue our series, and proceed to develop a monolithic Newton-multigrid thixoviscoplastic solver. The solver is based on Newton’s method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh’s elements, and proceed with outer blocks Gauss-Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton’s and multigrid methods. The adaptivity within discrete Newton’s method is based on the adaptive step-length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver’s update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid-driven cavity flow

FEM simulation of thixo-viscoplastic flow problems: Error analysis

This note is concerned with error analysis of FEM approximations for quasi-Newtonian modelling of thixo-viscoplastic, TVP, flow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization parameter, nonoptimal estimate for microstructure, and extra regularization requirement for velocity, is due to weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with higher order stabilized FEM. Furthermore, we showed the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical simulations of TVP flow problems in a 4:1 contraction configuration

Monolithic Finite Element Method for the simulation of thixo-viscoplastic flows

[EN] This note is concerned with the application of Finite Element Method (FEM) and NewtonMultigrid solver to simulate thixo-viscoplastic flows. The thixo-viscoplastic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyze the thixo-viscoplastic flow problem in a Lid-driven cavity configuration.The authors acknowledge the funding provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 446888252. Additionally, the authors acknowledge the financial grant provided by the Bundesministerium fr Wirtschaft und Energie aufgrund eines Beschlusses des Deutschen Bundestages through AiF-Forschungsvereinigung: Forschungs- Gesellschaft Verfahrens Technik e. V. - GVT under the IGF project number 20871 N. We would also like to gratefully acknowledge the support by LSIII and LiDO3 team at ITMC, TU Dortmund University, Germany.Begum, N.; Ouazzi, A.; Turek, S. (2022). Monolithic Finite Element Method for the simulation of thixo-viscoplastic flows. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 170-179. https://doi.org/10.4995/YIC2021.2021.12250OCS17017

Newton-multigrid FEM solver for the simulation of Quasi-Newtonian modeling of thixotropic flows

This paper is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers to simulate thixotropic flows using quasi-Newtonian modeling. The thixotropy phenomena are introduced to yield stress material by taking into consideration the in-ternal material microstructure using a structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropy throughout the structure parameter. Secondly, an evolution equation for the struc-ture parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. This is done simply by introduc-ing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The nonlinearity, related to the dependency of the diffusive term on the material parameters, is treated with generalized Newton’s method w.r.t. the Jacobian’s singularities having a global convergence property. The linearized systems inside the outer Newton loops are solved using the geometrical multigrid with a Vanka-like linear smoother taking into account a stable FEM approximation pair for velocity and pres-sure with discontinuous pressure and biquadratic velocity spaces. We analyze the application of using the quasi-Newtonian modeling approach for thixotropic flows, and the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of the thixotropic flows using manufactured solutions in a channel and the prototypical configuration of thixotropic flows in Couette device

FEM simulation of thixo-viscoplastic flow problems: error analysis

This note is concerned with the essential part of Finite Element Methods (FEM) approximation of error analysis for quasi-Newtonian modelling of thixo-viscoplastic (TVP) flow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization, nonoptimal estimate for microstructure, and extra regularity requirement for velocity, is due to the weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with stabilized higher order FEM. Furthermore, we show the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical solutions of TVP flow problems in a 4:1 contraction configuration

Finite element methods for the simulation of thixotropic flow problems

This note is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers for the simulation of thixotropic flow problems. The thixotropy phenomena are introduced into viscoplastic material by taking into ac-count the internal material micro structure using a scalar structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropic stress dependent on the structure parameter. Secondly, an evolution equation for the structure parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. Substantially, this is done sim-ply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The modified thixotropic viscoplastic stress w.r.t. the structure parameter is integrated in quasi-Newtonian manner into the generalized Navier-Stokes equations and the evolution equation for the structure parameter constitutes the main core of full set of modeling equations, which are creditable as the privilege answer to incorporate thixotropy phenomena. A fully coupled monolithic finite element approach has been exercised which manages the material internal micro structure parameter, ve-locity, and pressure fields simultaneously. The nonlinearity of the corresponding problem, related to the dependency of the diffusive stress on the material parameters and the non-linear structure parameter models on the other hand, is treated with generalized Newton’s method w.r.t. the Jacobian’s singularities having a global convergence property. The lin-earized systems inside the outer Newton loops form a typical saddle-point problem which is solved using a geometrical multigrid method with a Vanka-like smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous linear pressure and biquadratic velocity spaces. We examine the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of thixotropic viscoplastic flow problems in Couette device and in 4:1 contraction

Monolithic Newton-multigrid FEM for the simulation of thixotropic flow problems

This contribution is concerned with the application of Finite Element Method (FEM) and Newton-Multigrid solvers to simulate thixotropic flows. The thixotropic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyse the thixotropic viscoplastic flow problem in 4:1 contraction configuration

FEM simulations for thixo-viscoplastic flow problems: wellposedness results

In this contribution, we shall be concerned with the question of wellposedness of thixo viscoplastic flow problems in context of FEM approximations. We restrict our analysis to a quasi Newtonian modeling approach with the aim to set foundations for an efficient monolithic Newton multigrid solver. We present the wellposedness of viscoplastic subproblems and structure subproblems in parallel/independent fashion showing the possibility for a combined treatment. Then, we use the fixed point theorem for the coupled problem. For the numerical solutions, we choose 4:1 contraction config uration and use monolithic Newton-multigrid solver. We analyse the effect of taking into consideration thixotropic phenomena in viscoplastic material and opening up for more different coupling by inclusions of shear thickening and shear thinning behaviors for plastic viscosity and/or elastic behavior below the critical yield stress limit in more a general thixotropic models

FEM Modeling and Simulation of Thixo-viscoplastic Flow Problems

We are concerned, in this work, with Finite Element Method (FEM) for modeling and simulation of thixotropy in viscoplastic materials. We use a quasi-Newtonian approach to integrate the constitutive equation, which results in a new thixo-viscoplastic (TYP) generalized Navier-Stokes (N-8) equations. To solve the corresponding flow fields at once, we developed a FEM TYP solver based on monolithic Newton-multigrid method. The phenomenologi­cal process of competition of breakdown and buildup characteristics of thixotropic material is replicated throughout, localization and shear banding for Couette flow on one hand, and induction of more shear rejuvenation layers nearby walls for contraction flow on the other hand